General Constraints Mesh Adaptive Direct Search Filtering Algorithms

The paper presents and analysis the mesh adaptive direct search filter algorithm for the constrained optimization. General speaking, for solving unconstrained optimization, Newton and Quasi-Newton are common and basal algorithms. They judge whether exists a descent direction at this point by the information of vector gradient. However, on practical application, gradient is unavailable, either because it does not exist, or it cannot be accurately estimated due to noise in function or other reasons. Direct search is a method of solving nonsmooth optimization for nonlinear program without derivates. Generalized pattern search is a valuable direct method, but the application of nonsmooth analysis techniques showed its limitations due to the finite choice of directions. Mesh adaptive direct search extends the generalized pattern search class by allowing local exploration, called polling, in an asymptotically dense set of direction in the space of optimization variables.Mesh adaptive direct search filter method in the article is provided for the constrainted optimization. It is a derivative free direct method. Mesh adaptive direct search is different from linear search and region search, since it just satisfies the simple descent of function, instead of the sufficient descent. What is more, the algorithm adopts the same method as the filter algorithm to convert the constrained optimization to the multiobjective optimization, including the objective function and the violation constraint function, the measure of the constraint infeasibility. For the algorithm, it use the mesh adaptive direct search strategy, and use the filter to judge whether the each iteration is successful. Finally, the article show that the algorithm for the optimization exists the Clarke stationary point.